Do images submitted to a segmentation network after training need to be the same size as the training images?
Yes, you will need data that has the same dimensionality as your training set. Otherwise the model will either discard excess data, or will supplement missing data with a random number. This will likely lead to very poor results. 

I would suggest to rescale your images to have the same pixel density as those you used for training (250x250) rather than scaling your model. 

Methods for rescaling images and maintaining details are very well developed. Many exist and you can use existing tools such as Photoshop to change the pixel density of an image.